 A note on sumsets of subgroups in $\mathbb Z_p^*$  Derrick Hart Let $A$ be a multiplicative subgroup of $\mathbb Z_p^*$. Define the $k$fold sumset of $A$ to be $kA=\{x_1+\dots+x_k:x_i \in A,1\leq i\leq k\}$. We show that $6A\supseteq \mathbb Z_p^*$ for $A > p^{\frac {11}{23} +\epsilon}$. In addition, we extend a result of Shkredov to show that $2A\gg A^{\frac 85\epsilon}$ for $A\ll p^{\frac 59}$. Downloads: 2  
 derrick hart  fall asleep to this  derrick hart Derrick Hart is an American musician. You might know him from his "Songs From A Cross (The Sea)" EP on 12rec, a release in which Derrick shows great musical talent and a really fantastic sense for songwriting and melodies. With his "fall asleep to this" EP he goes a more experimental and meditative way. The five tracks are mainly based on vocals, even though there are no actual lyrics. "kontakt" and "when someone loves you no more" are performed only with his voice, captured with a contactmic,... Keywords: derrick hart; ambient; experimental; electronica Downloads: 4,104 
 Derrick Hart  Songs From A Cross (The Sea) [12rec.059]  Derrick Hart Following Nic Bommarito's "Harp Fragments", this is part II of our "America, the beautiful" series. I am following the enigmatic Derrick Hart for quite a while now. Equally rooted in the chromatics of USAmerican Folklore and offkey Pop music, Hart is blessed with a wonderful voice and musical talent that makes his compositions shine like nothing else I heard in a while. His four track "Songs from a cross (the sea)" EP is probably the shortest 12rec... Keywords: Derrick Hart; 12rec; 12rec.net; Indie; Pop; Folk; Singer Songwriter Downloads: 16,603 (2 reviews) 
 Sums and products in finite fields: an integral geometric viewpoint  Derrick Hart We prove that if $A \subset {\Bbb F}_q$ is such that $$A>q^{{1/2}+\frac{1}{2d}},$$ then $${\Bbb F}_q^{*} \subset dA^2=A^2+...+A^2 d \text{times},$$ where $$A^2=\{a \cdot a': a,a' \in A\},$$ and where ${\Bbb F}_q^{*}$ denotes the multiplicative group of the finite field ${\Bbb F}_q$. In particular, we cover ${\Bbb F}_q^{*}$ by $A^2+A^2$ if $A>q^{{3/4}}$. Furthermore, we prove that if $$A \ge C_{size}^{\frac{1}{d}}q^{{1/2}+\frac{1}{2(2d1)}},$$ then $$dA^2 \ge q \cdot \frac{C^2_{size}}{C^2... Downloads: 1  
 On sums and products in C[x]  Ernie Croot We show that under the assumption of a 24term version of Fermat's Last Theorem, there exists an absolute constant c > 0 such that if S is a set of n > n_0 positive integers satisfying S.S > n^2. In other words, we prove a weak form of the ErdosSzemeredi sumproduct conjecture, conditional on an extension of Fermat's Last Theorem. Unconditionally, we prove this theorem for when S is a set of n monic polynomials... Downloads: 2  
 Pinned distance sets, Wolff's exponent in finite fields and improved sumproduct estimates  Derrick Hart An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold $\alpha>0$ such that $\Delta(E) \gtrsim q$ whenever $E \gtrsim q^{\alpha}$, where $E \subset {\Bbb F}_q^d$, the $d$dimensional vector space over a finite field with $q$ elements (not necessarily prime). Here $\Delta(E)=\{{(x_1y_1)}^2+...+{(x_dy_d)}^2: x,y \in E\}$. The second listed author and Misha Rudnev established the threshold $\frac{d+1}{2}$, and the authors of this paper, Doowon Ko... Downloads: 2  
 kfold sums from a set with few products  Ernie Croot In the present paper we show that if A is a set of n real numbers, and the product set A.A has at most n^(1+c) elements, then the kfold sumset kA has at least n^(log(k/2)/2 log 2 + 1/2  f_k(c)) elements, where f_k(c) > 0 as c > 0. We believe that the methods in this paper might lead to a much stronger result; indeed, using a result of Trevor Wooley on Vinogradov's Mean Value Theorem and the TarryEscott Problem, we show that if A.A n^(Omega((k/log k)^(1/3))), for c small enough in terms o... Downloads: 1  
 Ubiquity of simplices in subsets of vector spaces over finite fields  Derrick Hart We prove that a sufficiently large subset of the $d$dimensional vector space over a finite field with $q$ elements, $ {\Bbb F}_q^d$, contains a copy of every $k$simplex. Fourier analytic methods, Kloosterman sums, and bootstrapping play an important role.  
 Fourier analysis and expanding phenomena in finite fields  Derrick Hart In this paper the authors study set expansion in finite fields. Fourier analytic proofs are given for several results recently obtained by Solymosi, Vinh and Vu using spectral graph theory. In addition, several generalizations of these results are given. In the case that $A$ is a subset of a prime field $\mathbb F_p$ of size less than $p^{1/2}$ it is shown that $\{a^2+b:a,b \in A\}\geq C A^{147/146}$, where $\cdot$ denotes the cardinality of the set and $C$ is an absolute constant. Downloads: 2  
 Averages over hyperplanes, sumproduct theory in vector spaces over finite fields and the ErdosFalconer distance conjecture  Derrick Hart We prove a pointwise and average bound for the number of incidences between points and hyperplanes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering ${\mathbb F}_q$, the finite field with q elements, by $A \cdot A+..... Downloads: 1  
 An analog of the FurstenbergKatznelsonWeiss theorem on triangles in sets of positive density in finite field geometries  David Covert We prove that if the cardinality of a subset of the 2dimensional vector space over a finite field with $q$ elements is $\ge \rho q^2$, with $\frac{1}{\sqrt{q}} Downloads: 5  
 Generalized incidence theorems, homogeneous forms, and sumproduct estimates in finite fields  David Covert In recent years, sumproduct estimates in Euclidean space and finite fields have been studied using a variety of combinatorial, number theoretic and analytic methods. Erdos type problems involving the distribution of distances, areas and volumes have also received much attention. In this paper we prove a relatively straightforward function version of an incidence results for points and planes previously established in \cite{HI07} and \cite{HIKR07}... Downloads: 4  
 Distance graphs in vector spaces over finite fields, coloring and pseudorandomness  Derrick Hart In this paper we systematically study various properties of the distance graph in ${\Bbb F}_q^d$, the $d$dimensional vector space over the finite field ${\Bbb F}_q$ with $q$ elements. In the process we compute the diameter of distance graphs and show that sufficiently large subsets of $d$dimensional vector spaces over finite fields contain every possible finite configurations. Downloads: 2  
 A FurstenbergKatznelsonWeiss type theorem on (d + 1)point configurations in sets of positive density in finite field geometries  David Covert We show that if $E \subset \mathbb{F}_q^d$, the $d$dimensional vector space over the finite field with $q$ elements, and $E \geq \rho q^d$, where $ q^{\frac{1}{2}}\ll \rho \leq 1$, then $E$ contains an isometric copy of at least $c \rho^{d1} q^{d+1 \choose 2}$ distinct $(d+1)$point configurations.  
 Pinned distance sets, ksimplices, Wolff's exponent in finite fields and sumproduct estimates  Jeremy Chapman An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold $\alpha>0$ such that $\Delta(E) \gtrsim q$ whenever $E \gtrsim q^{\alpha}$, where $E \subset {\Bbb F}_q^d$, the $d$dimensional vector space over a finite field with $q$ elements (not necessarily prime). Here $\Delta(E)=\{{(x_1y_1)}^2+...+{(x_dy_d)}^2: x,y \in E\}$. In two dimensions we improve the known exponent to $\tfrac{4}{3}$, consistent with the corresponding exponent in Euclidean sp... Downloads: 8  
